Question

Expand (x+y)^5 using the binomial theorem.

a. x^6+5xy^4+10x^3y^2+10x^2y^3+5xy^4
b.x^5+5xy^4+10x^3y^2+10x^2y^3+5xy^4
c. None of the above
d. Both a and b

Answer 1

The correct expansion of (x+y)^5 using the binomial theorem includes terms from x^5 to y^5. Both provided options (a and b) in the question are incorrect because neither lists all the terms in the expansion correctly. The correct answer is 'c. None of the above'.

Step-by-step explanation:

To expand (x+y)^5 using the binomial theorem, we apply the theorem which states that:

(a + b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2/2! + n(n-1)(n-2)a^(n-3)b^3/3! + ...

In this case, a is x, b is y, and n is 5. Therefore, the expansion of (x+y)^5 is:

• x^5 (the term with x raised to the power of 5)
• + 5x^4y (the next term with x raised to the power of 4 and y to the power of 1)
• + 10x^3y^2 (the term with x raised to the power of 3 and y to the power of 2)
• + 10x^2y^3 (the term with x raised to the power of 2 and y to the power of 3)
• + 5x^1y^4 (the term with x raised to the power of 1 and y to the power of 4)
• + y^5 (the term with y raised to the power of 5)

None of the provided options (a or b) fully match the correct expansion, so the answer is c. None of the above.